Aliases: CSU2(𝔽3)⋊1C4, C2.4(C4×S4), (C2×C4).1S4, (C2×Q8).4D6, Q8.2(C4×S3), (C4×Q8).4S3, Q8⋊Dic3.4C2, C22.10(C2×S4), C2.1(C4.S4), C2.1(Q8.D6), SL2(𝔽3).2(C2×C4), (C4×SL2(𝔽3)).1C2, (C2×CSU2(𝔽3)).2C2, (C2×SL2(𝔽3)).4C22, SmallGroup(192,947)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2 — Q8 — SL2(𝔽3) — C2×SL2(𝔽3) — C2×CSU2(𝔽3) — CSU2(𝔽3)⋊C4 |
| SL2(𝔽3) — CSU2(𝔽3)⋊C4 |
Generators and relations for CSU2(𝔽3)⋊C4
G = < a,b,c,d,e | a4=c3=e4=1, b2=d2=a2, bab-1=dbd-1=a-1, cac-1=ab, dad-1=a2b, ae=ea, cbc-1=a, be=eb, dcd-1=c-1, ce=ec, ede-1=a2d >
Subgroups: 223 in 67 conjugacy classes, 17 normal (15 characteristic)
C1, C2, C3, C4, C22, C6, C8, C2×C4, C2×C4, Q8, Q8, Dic3, C12, C2×C6, C42, C4⋊C4, C2×C8, Q16, C2×Q8, C2×Q8, SL2(𝔽3), C2×Dic3, C2×C12, C8⋊C4, Q8⋊C4, C4.Q8, C4×Q8, C4×Q8, C2×Q16, Dic3⋊C4, CSU2(𝔽3), C2×SL2(𝔽3), Q16⋊C4, Q8⋊Dic3, C4×SL2(𝔽3), C2×CSU2(𝔽3), CSU2(𝔽3)⋊C4
Quotients: C1, C2, C4, C22, S3, C2×C4, D6, C4×S3, S4, C2×S4, C4×S4, Q8.D6, C4.S4, CSU2(𝔽3)⋊C4
Character table of CSU2(𝔽3)⋊C4
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | |
| size | 1 | 1 | 1 | 1 | 8 | 2 | 2 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 8 | 8 | 8 | 8 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ4 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ5 | 1 | -1 | -1 | 1 | 1 | i | -i | 1 | -i | -1 | i | i | -1 | -i | 1 | -1 | 1 | -1 | i | -i | -1 | 1 | -i | -i | i | i | linear of order 4 |
| ρ6 | 1 | -1 | -1 | 1 | 1 | -i | i | 1 | i | -1 | -i | -i | -1 | i | 1 | -1 | 1 | -1 | -i | i | -1 | 1 | i | i | -i | -i | linear of order 4 |
| ρ7 | 1 | -1 | -1 | 1 | 1 | i | -i | 1 | -i | -1 | i | -i | 1 | i | -1 | -1 | 1 | -1 | -i | i | 1 | -1 | -i | -i | i | i | linear of order 4 |
| ρ8 | 1 | -1 | -1 | 1 | 1 | -i | i | 1 | i | -1 | -i | i | 1 | -i | -1 | -1 | 1 | -1 | i | -i | 1 | -1 | i | i | -i | -i | linear of order 4 |
| ρ9 | 2 | 2 | 2 | 2 | -1 | -2 | -2 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ10 | 2 | 2 | 2 | 2 | -1 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ11 | 2 | -2 | -2 | 2 | -1 | 2i | -2i | 2 | -2i | -2 | 2i | 0 | 0 | 0 | 0 | 1 | -1 | 1 | 0 | 0 | 0 | 0 | i | i | -i | -i | complex lifted from C4×S3 |
| ρ12 | 2 | -2 | -2 | 2 | -1 | -2i | 2i | 2 | 2i | -2 | -2i | 0 | 0 | 0 | 0 | 1 | -1 | 1 | 0 | 0 | 0 | 0 | -i | -i | i | i | complex lifted from C4×S3 |
| ρ13 | 3 | 3 | 3 | 3 | 0 | -3 | -3 | -1 | 1 | -1 | 1 | 1 | -1 | 1 | -1 | 0 | 0 | 0 | -1 | -1 | 1 | 1 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×S4 |
| ρ14 | 3 | 3 | 3 | 3 | 0 | 3 | 3 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | orthogonal lifted from S4 |
| ρ15 | 3 | 3 | 3 | 3 | 0 | -3 | -3 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | 0 | 0 | 0 | 1 | 1 | -1 | -1 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×S4 |
| ρ16 | 3 | 3 | 3 | 3 | 0 | 3 | 3 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | orthogonal lifted from S4 |
| ρ17 | 3 | -3 | -3 | 3 | 0 | 3i | -3i | -1 | i | 1 | -i | -i | 1 | i | -1 | 0 | 0 | 0 | i | -i | -1 | 1 | 0 | 0 | 0 | 0 | complex lifted from C4×S4 |
| ρ18 | 3 | -3 | -3 | 3 | 0 | -3i | 3i | -1 | -i | 1 | i | -i | -1 | i | 1 | 0 | 0 | 0 | i | -i | 1 | -1 | 0 | 0 | 0 | 0 | complex lifted from C4×S4 |
| ρ19 | 3 | -3 | -3 | 3 | 0 | 3i | -3i | -1 | i | 1 | -i | i | -1 | -i | 1 | 0 | 0 | 0 | -i | i | 1 | -1 | 0 | 0 | 0 | 0 | complex lifted from C4×S4 |
| ρ20 | 3 | -3 | -3 | 3 | 0 | -3i | 3i | -1 | -i | 1 | i | i | 1 | -i | -1 | 0 | 0 | 0 | -i | i | -1 | 1 | 0 | 0 | 0 | 0 | complex lifted from C4×S4 |
| ρ21 | 4 | 4 | -4 | -4 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C4.S4, Schur index 2 |
| ρ22 | 4 | -4 | 4 | -4 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8.D6, Schur index 2 |
| ρ23 | 4 | 4 | -4 | -4 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | -1 | 0 | 0 | 0 | 0 | √3 | -√3 | -√3 | √3 | symplectic lifted from C4.S4, Schur index 2 |
| ρ24 | 4 | 4 | -4 | -4 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | -1 | 0 | 0 | 0 | 0 | -√3 | √3 | √3 | -√3 | symplectic lifted from C4.S4, Schur index 2 |
| ρ25 | 4 | -4 | 4 | -4 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | 1 | 0 | 0 | 0 | 0 | -√-3 | √-3 | -√-3 | √-3 | complex lifted from Q8.D6 |
| ρ26 | 4 | -4 | 4 | -4 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | 1 | 0 | 0 | 0 | 0 | √-3 | -√-3 | √-3 | -√-3 | complex lifted from Q8.D6 |
►On 64 points
Generators in S64
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)(25 26 27 28)(29 30 31 32)(33 34 35 36)(37 38 39 40)(41 42 43 44)(45 46 47 48)(49 50 51 52)(53 54 55 56)(57 58 59 60)(61 62 63 64)
(1 11 3 9)(2 10 4 12)(5 63 7 61)(6 62 8 64)(13 18 15 20)(14 17 16 19)(21 26 23 28)(22 25 24 27)(29 34 31 36)(30 33 32 35)(37 43 39 41)(38 42 40 44)(45 51 47 49)(46 50 48 52)(53 59 55 57)(54 58 56 60)
(2 11 10)(4 9 12)(5 8 62)(6 64 7)(13 19 18)(15 17 20)(21 27 26)(23 25 28)(29 35 34)(31 33 36)(38 43 42)(40 41 44)(46 51 50)(48 49 52)(54 59 58)(56 57 60)
(1 37 3 39)(2 41 4 43)(5 29 7 31)(6 36 8 34)(9 38 11 40)(10 44 12 42)(13 49 15 51)(14 47 16 45)(17 46 19 48)(18 52 20 50)(21 57 23 59)(22 55 24 53)(25 54 27 56)(26 60 28 58)(30 63 32 61)(33 62 35 64)
(1 32 16 24)(2 29 13 21)(3 30 14 22)(4 31 15 23)(5 49 59 41)(6 50 60 42)(7 51 57 43)(8 52 58 44)(9 33 17 25)(10 34 18 26)(11 35 19 27)(12 36 20 28)(37 63 45 55)(38 64 46 56)(39 61 47 53)(40 62 48 54)
G:=sub<Sym(64)| (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32)(33,34,35,36)(37,38,39,40)(41,42,43,44)(45,46,47,48)(49,50,51,52)(53,54,55,56)(57,58,59,60)(61,62,63,64), (1,11,3,9)(2,10,4,12)(5,63,7,61)(6,62,8,64)(13,18,15,20)(14,17,16,19)(21,26,23,28)(22,25,24,27)(29,34,31,36)(30,33,32,35)(37,43,39,41)(38,42,40,44)(45,51,47,49)(46,50,48,52)(53,59,55,57)(54,58,56,60), (2,11,10)(4,9,12)(5,8,62)(6,64,7)(13,19,18)(15,17,20)(21,27,26)(23,25,28)(29,35,34)(31,33,36)(38,43,42)(40,41,44)(46,51,50)(48,49,52)(54,59,58)(56,57,60), (1,37,3,39)(2,41,4,43)(5,29,7,31)(6,36,8,34)(9,38,11,40)(10,44,12,42)(13,49,15,51)(14,47,16,45)(17,46,19,48)(18,52,20,50)(21,57,23,59)(22,55,24,53)(25,54,27,56)(26,60,28,58)(30,63,32,61)(33,62,35,64), (1,32,16,24)(2,29,13,21)(3,30,14,22)(4,31,15,23)(5,49,59,41)(6,50,60,42)(7,51,57,43)(8,52,58,44)(9,33,17,25)(10,34,18,26)(11,35,19,27)(12,36,20,28)(37,63,45,55)(38,64,46,56)(39,61,47,53)(40,62,48,54)>;
G:=Group( (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32)(33,34,35,36)(37,38,39,40)(41,42,43,44)(45,46,47,48)(49,50,51,52)(53,54,55,56)(57,58,59,60)(61,62,63,64), (1,11,3,9)(2,10,4,12)(5,63,7,61)(6,62,8,64)(13,18,15,20)(14,17,16,19)(21,26,23,28)(22,25,24,27)(29,34,31,36)(30,33,32,35)(37,43,39,41)(38,42,40,44)(45,51,47,49)(46,50,48,52)(53,59,55,57)(54,58,56,60), (2,11,10)(4,9,12)(5,8,62)(6,64,7)(13,19,18)(15,17,20)(21,27,26)(23,25,28)(29,35,34)(31,33,36)(38,43,42)(40,41,44)(46,51,50)(48,49,52)(54,59,58)(56,57,60), (1,37,3,39)(2,41,4,43)(5,29,7,31)(6,36,8,34)(9,38,11,40)(10,44,12,42)(13,49,15,51)(14,47,16,45)(17,46,19,48)(18,52,20,50)(21,57,23,59)(22,55,24,53)(25,54,27,56)(26,60,28,58)(30,63,32,61)(33,62,35,64), (1,32,16,24)(2,29,13,21)(3,30,14,22)(4,31,15,23)(5,49,59,41)(6,50,60,42)(7,51,57,43)(8,52,58,44)(9,33,17,25)(10,34,18,26)(11,35,19,27)(12,36,20,28)(37,63,45,55)(38,64,46,56)(39,61,47,53)(40,62,48,54) );
G=PermutationGroup([[(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24),(25,26,27,28),(29,30,31,32),(33,34,35,36),(37,38,39,40),(41,42,43,44),(45,46,47,48),(49,50,51,52),(53,54,55,56),(57,58,59,60),(61,62,63,64)], [(1,11,3,9),(2,10,4,12),(5,63,7,61),(6,62,8,64),(13,18,15,20),(14,17,16,19),(21,26,23,28),(22,25,24,27),(29,34,31,36),(30,33,32,35),(37,43,39,41),(38,42,40,44),(45,51,47,49),(46,50,48,52),(53,59,55,57),(54,58,56,60)], [(2,11,10),(4,9,12),(5,8,62),(6,64,7),(13,19,18),(15,17,20),(21,27,26),(23,25,28),(29,35,34),(31,33,36),(38,43,42),(40,41,44),(46,51,50),(48,49,52),(54,59,58),(56,57,60)], [(1,37,3,39),(2,41,4,43),(5,29,7,31),(6,36,8,34),(9,38,11,40),(10,44,12,42),(13,49,15,51),(14,47,16,45),(17,46,19,48),(18,52,20,50),(21,57,23,59),(22,55,24,53),(25,54,27,56),(26,60,28,58),(30,63,32,61),(33,62,35,64)], [(1,32,16,24),(2,29,13,21),(3,30,14,22),(4,31,15,23),(5,49,59,41),(6,50,60,42),(7,51,57,43),(8,52,58,44),(9,33,17,25),(10,34,18,26),(11,35,19,27),(12,36,20,28),(37,63,45,55),(38,64,46,56),(39,61,47,53),(40,62,48,54)]])
Matrix representation of CSU2(𝔽3)⋊C4 ►in GL7(𝔽73)
| 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 72 | 72 | 72 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 72 | 72 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 72 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 72 | 72 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 72 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 32 | 46 | 27 | 46 |
| 0 | 0 | 0 | 46 | 27 | 32 | 46 |
| 0 | 0 | 0 | 27 | 32 | 46 | 46 |
| 0 | 0 | 0 | 46 | 46 | 46 | 41 |
| 27 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 27 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 27 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 30 | 43 | 30 |
| 0 | 0 | 0 | 43 | 0 | 30 | 30 |
| 0 | 0 | 0 | 30 | 43 | 0 | 30 |
| 0 | 0 | 0 | 43 | 43 | 43 | 0 |
G:=sub<GL(7,GF(73))| [0,72,1,0,0,0,0,0,72,0,0,0,0,0,1,72,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,1,0,0,0,0,0,0,0,1,0,0],[0,1,72,0,0,0,0,1,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,1,0],[1,0,72,0,0,0,0,0,0,72,0,0,0,0,0,1,72,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,72,0,0,0,0,0,0,0,72,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,32,46,27,46,0,0,0,46,27,32,46,0,0,0,27,32,46,46,0,0,0,46,46,46,41],[27,0,0,0,0,0,0,0,27,0,0,0,0,0,0,0,27,0,0,0,0,0,0,0,0,43,30,43,0,0,0,30,0,43,43,0,0,0,43,30,0,43,0,0,0,30,30,30,0] >;
CSU2(𝔽3)⋊C4 in GAP, Magma, Sage, TeX
{\rm CSU}_2({\mathbb F}_3)\rtimes C_4 % in TeX
G:=Group("CSU(2,3):C4"); // GroupNames label
G:=SmallGroup(192,947);
// by ID
G=gap.SmallGroup(192,947);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,2,-2,672,1373,36,451,1684,655,172,1013,404,285,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=c^3=e^4=1,b^2=d^2=a^2,b*a*b^-1=d*b*d^-1=a^-1,c*a*c^-1=a*b,d*a*d^-1=a^2*b,a*e=e*a,c*b*c^-1=a,b*e=e*b,d*c*d^-1=c^-1,c*e=e*c,e*d*e^-1=a^2*d>;
// generators/relations
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